Miss Minerva challenges her class to guess her secret positive integer. She gives them these clues: The sum of the digits of this three-digit number is 14. The tens digit is the cube of a prime number. The units digit is one-fifth of the hundreds digit. What is Miss Minerva's number?
Solution: The only cube of a prime number that is less than 10 is $2^3=8$, so the tens digit is 8, which means the hundreds digit and units digit sum to $14-8=6$. We're given that the units digit is one-fifth of the hundreds digit, so the units digit is $1$ and the hundreds digit is $5$. The desired number is $\boxed{581}$.